Phase Space & SM: Universe Defined?

[Jimster41]

Is the phase space of the universe, including it's partitioning, defined and calculated from the the SM?

 

[ohwilleke]

This question is ambiguous on its face without providing more context. There is not one unique "phase space" of the universe. Every phase space is a phase space of some theory or construct that creates a quantity that translates real world events into phase space coordinates. There are lots of completely different theories that create completely different phase spaces and I can't tell from your question which one you mean.

This said, pretty much anything on the scale of the universe is not adequately described using the Standard Model, because the Standard Model does not include General Relativity or any theory of gravity, and at the scale of the universe, unlike the scale at which typical experiments testing the SM are conducted, General Relativity has multiple material impacts on the behavior of the system that can not be ignored as they are in typical HEP experiments. Pretty much any question related to the entire universe is beyond the domain of applicability of the SM.

 

[Jimster41]

A bunch of n coins has a phase space that is defined by the fact coins can have one of two values, and there are n of them. They have a partition function, and a notion of entropy and ergodicity defined by way those featues distribute across their phase or configuration space.

A physical theory that described all the things that can exist and the ways those can exist would have a phase space, entropy and ergodicity defined in a similar way?

Is that correct?

 

[ohwilleke]

Thank you for clarifying your question which I now understand much better.

The short answer is no. The SM is not such a theory as you have defined it.

Moreover, I'm not sure that this definition can be imposed, even in principle, on any quantum system, even if some of the shortcomings peculiar to the SM as of July 30, 2015 like its failure to address dark matter, dark energy and quantum gravity that would be necessary to make the SM as "Theory of Everything" (i.e. TOE) were resolved at some point in the future.

Basically, as I understand your question, the meat of what you are getting at is whether the SM, standing alone, creates not a Newtonian clockwork universe perhaps, but a post-Newtonian board game universe where you can know all of the pieces on the board at anyone time and use "dice" to advance each piece to the right place on the board on the next turn. For a lot of purposes, that is an approximation of reality that is adequate. But, the reality is more difficult for fundamental reasons of what quantum physics means and not just because we seem to have misplaced a couple of big boxes of pieces to put on the board and a page or two of the rules of the game.

Anyway, point by point:

1.The jury is decidedly out on the question of whether the SM describes all things that can exist, but the answer is likely to be no.

The SM describes lots of things that exist right now, but it doesn't describe anything that is plausibly the source of dark matter or dark energy phenomena which is the lion's share of the matter-energy of the universe so far as we know. It is likewise the case that it probably omits some stuff that can exist only at GUT scale energies and probably existed in the universe in the past (e.g. some sort of inflaton and possibly something intermediate between pure energy and baryogenesis and leptogenesis), but a lot of that stuff may have once existed but probably doesn't actually exist right now and probably won't exist ever in the future unless somebody artificially creates a truly amazing, galactic civilization millions of years more advanced than our own, artificially created experiment to do so. Your question is ambiguous with regard to the time frame in which a theory must qualify.

2.The SM describes all of the properties that a SM particle can have other than the gravitational fields that they create. But, at any given time lots of those particles are in unobserved transitions and have indeterminate quantum states so it is not at all obvious that a partition function could be adequately applied to many of them (or even any of them) with perfect precision.

In other words, while it might be possible to partition SM particles to some extent in some manner, it is not possible to do so with arbitrary precision, and with respect to all of their properties of every particle at the same time. At a minimum, you would need to define your partition function in a manner more more subtle than what you would need to do in a classical Newtonian clock work universe, and I don't believe that anyone has been clever enough to come up with such a partition function yet, not that it wouldn't be worth the time of someone very clever to try. There is certainly no obvious way to do so.

3. In principle a particle's SM properties should be sufficient to describe its contribution to stress-energy tensor that gives rise to its contribution to a gravitational field, as the SM is used to determine observable quantities that go into each of the nine stress-energy tensor boxes. But, of course, the SM does not describe the properties of gravitons in quantum gravity theories if they, rather than GR, are accurate, or the properties of anything else that contributes to Dark Matter or Dark Energy. GR also isn't really equipped to be applied directly to quantum point particles at all. For example, it is not known how the equations of GR would be applied in the case of a single electron, because a point particle with finite mass is a singularity. But, it might be the case that the proper way to integrate a single electron into GR is to treat it as if it is smeared over the area occupied by its amplitude field or its Compton radius sphere, which would prevent it from being a singularity. This is a fundamental unsolved problem of physics.

4. Entropy has a definition in physics, but this definition is not generally considered to be part of the Standard Model. There are lots of good operationally viable ways to define entropy in systems that are large enough to be well defined by classical physics that allow us to estimate it in actual physical systems, and like coordinate systems in GR, the key it to be consistent in your definitions, rather than to embrace a preferred definition of a particular quantitative description of the entropy of a system. Many conventional ways of operationally defining entropy in systems at a classical scale are not well suited to being applied to indeterminate quantum states (e.g. photons en route from a source to a target). I personally don't know just how you would go about defining entropy in some very basic situations like that one, but perhaps there is such a definition that I am not aware of. For example, entropy is normally considered a point in time measurement in classical physics, but in the Standard Model, at least some of properties of particles in a system used to determine its microstates for purposes of entropy and the application of the thermodynamic ergotic hypothesis at a sufficiently precisely defined point in time are arbitrarily uncertain.

I suspect that these issues could be overcome with a judicious exercise of calculus and the right definition, but the definition is beyond the scope of the SM itself, and I don't believe that there is any consensus off the self way to answer this question.

5.The thermodynamic ergotic hypothesis is probably true in systems described by the Standard Model but I don't know whether or not this has ever been rigorously proved.

 

[bapowell]

4. Entropy has a definition in physics, but this definition is not generally considered to be part of the Standard Model. There are lots of good operationally viable ways to define entropy in systems that are large enough to be well defined by classical physics that allow us to estimate it in actual physical systems, and like coordinate systems in GR, the key it to be consistent in your definitions, rather than to embrace a preferred definition of a particular quantitative description of the entropy of a system. Many conventional ways of operationally defining entropy in systems at a classical scale are not well suited to being applied to indeterminate quantum states (e.g. photons en route from a source to a target). I personally don't know just how you would go about defining entropy in some very basic situations like that one, but perhaps there is such a definition that I am not aware of. For example, entropy is normally considered a point in time measurement in classical physics, but in the Standard Model, at least some of properties of particles in a system used to determine its microstates for purposes of entropy and the application of the thermodynamic ergotic hypothesis at a sufficiently precisely defined point in time are arbitrarily uncertain.

Quantum statistical mechanics is a well-established field: https://en.wikipedia.org/wiki/Von_Neumann_entropy

 

[ohwilleke]

Quantum statistical mechanics is a well-established field: https://en.wikipedia.org/wiki/Von_Neumann_entropy

Fair enough. That clever person has already come along. (Though it still isn't part of the SM per se). I'd also note that the link above also makes clear that a partition function is a classical concept and that an analogous concept but not an identical one must be used in quantum statistical mechanics, and that there is dispute over exactly how to correctly define entropy with some potential flaws identified in Von Neumann entropy, and with the definition used in Von Neumann entropy being dependent upon being used at the time that an actual measurement is conducted, rather than over a continuous period of time for all the space in the universe.

 

[Jimster41]

assume n "hidden" QM variables, with some bounded (but unknown) set of eigenvectors and eigenvalues.
assume some SM (or GUT) was completed and had described all classical observations

Would proof via ergodic decomposition be coherent?
If space QM(hidden) is ergodic and space CM(observable) is ergodic, then the combined space QM⊗CM is ergodic.

 

[ohwilleke]

Naively, it would seem to be, but those are some big "ifs" and the issue is subtle enough that I wouldn't trust my first impressions without trying to be more rigorous.

 

[Jimster41]

That's more than fair. I was imagining that such a setup might exposes a handle on the information or entropy flow (which is it?) of the classical second law via Crook's Fluctuation Theorem, if it can be seen to sit between the two spaces, as some sort of bit pipe.

I need to really try to read into that Weismann paper.
http://arxiv.org/pdf/quant-ph/0605031v1.pdfRegardless, I feel like I finally made some progress getting through Susskind QM, by trying to grapple with this cartoon while also reading ch6 for literally the sixteenth time.

 

[Jimster41]

So looking at say Garrett Lisi's E₈ Lie Group, as an example, is the idea of a phase space map accessible? Or at least some characteristic features of it? Like its specific ergodicity (aren't there different types?)

I can picture the principle of sizing and structuring the phase space of some coins, maybe even some die, or some spinors (barely), but how does one even begin to think about the size and structure phase space of an algebra...? Dimensionality is specified, but what is the base, then how is structure, and reduction from raw geometric combination even discussed?